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Theorem pmtrsn 18297
Description: The value of the transposition generator function for a singleton is empty, i.e. there is no transposition for a singleton. This also holds for 𝐴 ∉ V, i.e. for the empty set {𝐴} = ∅ resulting in (pmTrsp‘∅) = ∅. (Contributed by AV, 6-Aug-2019.)
Assertion
Ref Expression
pmtrsn (pmTrsp‘{𝐴}) = ∅

Proof of Theorem pmtrsn
Dummy variables 𝑝 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 5131 . . 3 {𝐴} ∈ V
2 eqid 2825 . . . 4 (pmTrsp‘{𝐴}) = (pmTrsp‘{𝐴})
32pmtrfval 18227 . . 3 ({𝐴} ∈ V → (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
41, 3ax-mp 5 . 2 (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
5 eqid 2825 . . . . 5 (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
65dmmpt 5875 . . . 4 dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
7 2on0 7841 . . . . . . . . 9 2o ≠ ∅
8 ensymb 8276 . . . . . . . . . 10 (∅ ≈ 2o ↔ 2o ≈ ∅)
9 en0 8291 . . . . . . . . . 10 (2o ≈ ∅ ↔ 2o = ∅)
108, 9bitri 267 . . . . . . . . 9 (∅ ≈ 2o ↔ 2o = ∅)
117, 10nemtbir 3094 . . . . . . . 8 ¬ ∅ ≈ 2o
12 snnen2o 8424 . . . . . . . 8 ¬ {𝐴} ≈ 2o
13 0ex 5016 . . . . . . . . 9 ∅ ∈ V
14 breq1 4878 . . . . . . . . . 10 (𝑦 = ∅ → (𝑦 ≈ 2o ↔ ∅ ≈ 2o))
1514notbid 310 . . . . . . . . 9 (𝑦 = ∅ → (¬ 𝑦 ≈ 2o ↔ ¬ ∅ ≈ 2o))
16 breq1 4878 . . . . . . . . . 10 (𝑦 = {𝐴} → (𝑦 ≈ 2o ↔ {𝐴} ≈ 2o))
1716notbid 310 . . . . . . . . 9 (𝑦 = {𝐴} → (¬ 𝑦 ≈ 2o ↔ ¬ {𝐴} ≈ 2o))
1813, 1, 15, 17ralpr 4459 . . . . . . . 8 (∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o ↔ (¬ ∅ ≈ 2o ∧ ¬ {𝐴} ≈ 2o))
1911, 12, 18mpbir2an 702 . . . . . . 7 𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o
20 pwsn 4652 . . . . . . . 8 𝒫 {𝐴} = {∅, {𝐴}}
2120raleqi 3354 . . . . . . 7 (∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o ↔ ∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o)
2219, 21mpbir 223 . . . . . 6 𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o
23 rabeq0 4188 . . . . . 6 ({𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅ ↔ ∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o)
2422, 23mpbir 223 . . . . 5 {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅
25 rabeq 3405 . . . . 5 ({𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅ → {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V})
2624, 25ax-mp 5 . . . 4 {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
27 rab0 4187 . . . 4 {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = ∅
286, 26, 273eqtri 2853 . . 3 dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅
29 funmpt 6165 . . . . 5 Fun (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
30 funrel 6144 . . . . 5 (Fun (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
3129, 30ax-mp 5 . . . 4 Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
32 reldm0 5579 . . . 4 (Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅))
3331, 32ax-mp 5 . . 3 ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅)
3428, 33mpbir 223 . 2 (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅
354, 34eqtri 2849 1 (pmTrsp‘{𝐴}) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198   = wceq 1656  wcel 2164  wral 3117  {crab 3121  Vcvv 3414  cdif 3795  c0 4146  ifcif 4308  𝒫 cpw 4380  {csn 4399  {cpr 4401   cuni 4660   class class class wbr 4875  cmpt 4954  dom cdm 5346  Rel wrel 5351  Fun wfun 6121  cfv 6127  2oc2o 7825  cen 8225  pmTrspcpmtr 18218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-om 7332  df-1o 7831  df-2o 7832  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-pmtr 18219
This theorem is referenced by:  psgnsn  18298
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